Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle CBD = 3x - 14$, and $ m \angle ABC = 4x - 22$, find $m\angle ABC$. $B$ $A$ $D$ $C$
Explanation: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {4x - 22} + {3x - 14} = {90}$ Combine like terms: $ 7x - 36 = 90$ Add $36$ to both sides: $ 7x = 126$ Divide both sides by $7$ to find $x$ $ x = 18$ Substitute $18$ for $x$ in the expression that was given for $m\angle ABC$ $ m\angle ABC = 4({18}) - 22$ Simplify: $ {m\angle ABC = 72 - 22}$ So ${m\angle ABC = 50}$.